Efficient class of estimators for finite population mean using auxiliary attribute in stratified random sampling

The aim of this paper is to develop more effective methods for estimating population means in sample surveys using auxiliary attributes. To achieve this goal, we introduce a modified version of the estimators proposed by Koyuncu (2013b) and Shahzad et al. (2019), as well as a new class of estimators. We derive expressions for the bias and mean squared error of these new estimators up to the first degree of approximation. Our results show that the suggested classes of estimators perform better than other existing methods, with the lowest mean squared error under optimal conditions. We also conduct an empirical investigation to support our findings.

Notations. Consider a population size N unit, is divided into L strata units hth stratum containing N h units, where h = 1, 2,…, L such that L h=1 N h = N . A simple random sample of size n h is drawn without replacement the hth stratum such that L h=1 n h = n . Let y hi , φ hi be observed value of study variable y and the auxiliary attribute φ on the ith unit of the hth stratum, respectively, where i = 1, 2, ..., N h and h = 1, 2, ..., L.
Further, let y h = and ρ yφh = s yφh s yh s φh are the hth sample covariance and point bi-serial correlation and S yφh = N h i=1 (yhi−Yh)(φhi−Ph) (N h −1) and ρ yφh = S yφh S yh S φh are population covariance and point bi-serial correlation between the study variable y and the auxiliary attribute φ , respectively.
To derive the bias and mean squared error (MSE) of the estimators, we write such that E(e o ) = E(e 1 ) = 0 and where p st = L h=1 W h p h such that E p st = P = L h=1 W h P h and γ h = N h −n h n h N h .
Reviewing some existing estimators. The conventional unbiased estimators for population mean Y of the study variable y under stratified random sampling is given by The variance/MSE of the estimator t 0 is given by The ratio estimator for population mean Y using auxiliary attribute φ in stratified random sampling due to Naik and Gupta 4 is given by The MSE of the estimator t 1 up to the first degree of approximation (fda), is given by The stratified version of ordinary product estimator for population mean Y is defined by The MSE of t 2 to the fda, is given by The usual regression estimator for Y is given by where b st is the sample regression coefficient of y on φ.
To the fda the MSE of t 3 is given by  4 . The MSE of the estimator t 4 to the fda is given by (10) is minimized for Therefore, the resulting minimum MSE of t 4 is given by Using information on auxiliary attribute φ , Sharma and Singh 3 proposed the following exponential type estimators for Y as where α being a suitable chosen constant.
To the fda the MSEs of the estimators t 1e, t 2e and t αe, are respectively given by The MSE(t αe ) is minimum when This yields the minimum MSE of t αe is given by (13) t 1e = y st exp P − p st P + p st , (ratio type exponential estimator) (14) t 2e = y st exp p st − P p st + P , (product-type exponential estimator)   (24) is minimized for Therefore, the minimum MSE of t 5 is given by Putting a st = 1 and b st = NP in (24), we get the MSE of t 6 to the fda is given by (27) is minimum when Therefore, the minimum MSE of t 6 is given by Koyuncu 10 and Shahzad et al. 11 proposed the following class of estimators for Y as where (w 1 , w 2 , γ ) are suitable chosen constants and (a st , b st ) are same as defined earlier.
To the fda, the MSE of t 7 is given by .
(30)  (31) is minimized for Therefore, the minimum MSE of t 7 is given by In this paper we have suggested a class of estimators for population mean Y of the study variable y using auxiliary attribute φ . Expressions of bias and MSE of the proposed class of estimators are obtained up to terms of order 0 (n −1 ).
We have obtained the optimum condition under which the MSE of the proposed class of estimators is minimum. We have derived the conditions under which the suggested class of estimators is more efficient than the conventional estimator and the estimators due to Naik and Gupta 4 , Koyuncu 9 , Sharma and Singh 3 and Shahzad et al. 11 . Numerical illustration is given in support of the proposed study.

Suggested class of estimators
We note that the exponent part of (30) is obtained on using the transformation a st p st + b st such that The MSE t 7(m) at (36) is minimized for Therefore, the minimum MSE of t 7(m) is given by Now we can conclude this as a theorem given below.

An alternative class of estimators
We propose another class of estimators for population mean as Y as  (41) is minimized for Substitution of (42) in (41) provides the minimum MSE of t 8 is given by Now we have the following theorem.
Further from (12) Therefore we can say that the proposed class of estimators t 8 is more efficient than the estimators t 4, t 5 , t 6 , t 7 and t 7(m) as long as the conditions (55), (56), (57), (58), and (59) respectively are satisfied.

Numerical illustration
To judge the merits of the suggested class of estimators t 8 over other existing estimators, we have computed the percent relative efficiency (PRE) of different estimators with respect to usual unbiased estimator y st by using the following formulae: × 100, The summary statistics of the data are given in Table 1. We applied Neyman 23 allocation for allocating the samples to various strata 24 . Source: Koyuncu 9 .
It is observed from Table 2 that the estimators t 1 and t 1e are more efficient than the usual unbiased estimator y (which does not utilize the auxiliary attribute). The product estimator t 2 and product-type exponential estimator t 2(e) perform poor than y (due to positive correlation between y and φ). The t 3 is more efficient than t 1 , t 1e , t 2 and t 2e . The performance of the estimators (t 4 , t 5 , t 6 ) are almost same but marginally better than estimators t 1 , t 1e , t 2 , t 2e and t 3 .
× 100,  Table 2. PREs values of different estimators of Y with respect to y.  2 also demonstrates that the proposed estimator t 8 with δ = −1, η = 1, a st = C φ(st) , b st = NP has the largest PRE(= 1.49E+11) followed by t 7(m) with = −1, a st = C φ(st) , b st = NP . It is further observed that the proposed classes of estimators t 7(m) and t 8 are always better than the classes of estimators t 1 4 , t 1e , t 2 and t 2e , t 3 (difference estimator), t 4 9 , t 5 3 , t 6 11 , t 7 10 for all choices of (a st , b st ) . The proposed class of estimators t 8 is the best among the estimators closed in Table 2.

Different values of scalars Estimators
Thus, our recommendation is to use the suggested class of estimators t 7(m) and t 8 in practice.

Conclusion
In this article, we propose two classes of estimators for estimating the population mean Y of the study variable y using information on an auxiliary attribute (φ) . The suggested classes of estimators are wide-ranging. The biases and mean squared errors of the proposed classes of estimators t 7(m) and t 8 are derived up to the first degree of approximation. The optimum estimators in the classes of estimators t 7(m) and t 8 are investigated using the minimum mean squared error formulae. An empirical study is conducted to evaluate the efficiency of the proposed classes of estimators t 7(m) and t 8 and the findings are presented in Table 2. The results of Table 2 demonstrate that the suggested classes of estimators t 7(m) and t 8 are more efficient than the recently developed classes of estimators t 4 , t 5 ,t 6 , t 7 and t 3 by Koyuncu 9 , Sharma and Singh 3 , Shahzad et al. 11 , Koyuncu 10 , and the difference estimator, with a considerable gain in efficiency. Therefore, we conclude that the proposed classes of estimators and are justified and can be used in practice. One potential direction for future research is the application of advanced statistical techniques, such as machine learning and artificial intelligence, can be explored to improve the accuracy and efficiency of the estimators. These techniques can also help in identifying relevant auxiliary variables for improving the estimation process. The impact of various sampling designs on the estimation process can be investigated. For example, the effect of unequal sample sizes in different strata, non-response rates, and measurement errors on the accuracy and efficiency of the estimators can be studied. Finally, the extension of the current research to other types of population parameters, such as variance and quantiles, can also be explored. This can lead to the development of new classes of estimators and further improve the accuracy and efficiency of the estimation process.